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2 • a choice of characteristic class c1(ξ) ∈ E (X) for complex line bun- 1 dles ξ → X such that, for the canonical line bundle γ1 → CP , the 2 1 ∼ 0 isomorphism E (CP ) = E carries c1(γ1)e to 1;

2i • a family of characteristice classes ci(ξ) ∈ E (X) for complex vector bundles ξ → X satisfying the above formula for c1(γ1) and such that the Cartan formula for Whitney sums holds;e or

• a natural transformation MU → E of multiplicative cohomology theo- ries.

In the third case, the natural transformation MU 2i(X) → E2i(X) allows us MU E to push forward the characteristic classes ci (ξ) to classes ci (ξ). Such a map MU → E is called a complex orientation of E, and as a result MU plays a fundamental role in the theory of Chern classes. Moving from the homotopy category to the point-set level, the spectrum MU representing complex bordism is also one of the best known examples of a spectrum with a multiplication which is associative and commutative up to all higher coherences (an E∞ ring structure). If we know that E is also equipped with an E∞ ring structure, it is natural to ask whether a complex orientation MU → E can be lifted to a map respecting this E∞ ring structure (an E∞ orientation), and what data is necessary to describe this. This is a stubborn problem and in prominent cases the answer is unknown, such as when E is a Lubin–Tate spectrum. For any complex orientation of E, there is a formal group law G expressing the first Chern class of a tensor product of two complex line bundles: we have ′ ′′ ′ ′′ c1(ξ ⊗C ξ )= c1(ξ )+G c1(ξ ) ∗ for an associative, commutative, and unital power series x +G y ∈ E Jx, yK. Ando gave a necessary and sufficient condition for an orientation MU → E to be an H∞ orientation [And92], a weaker structure than an E∞ orientation

2 which can be described as a natural transformation that respects geometric power operations. Ando’s condition was that a certain natural power opera- tion Ψ with source E∗(X) should act as the canonical Lubin isogeny on the coordinate ring E∗(CP∞) of the formal group law G (see also [AHS04, §4.3]). In general, this is stronger than the data of a complex orientation alone [JN10], and very few E∞ ring spectra are known to admit H∞ orientations. (Ando also showed that Lubin–Tate spectra associated to the Honda formal group law have unique H∞ orientations; in recent work, Zhu has generalized this to all of the Lubin–Tate spectra [Zhu17].)

However, it is the case that the rationalization MUQ is universal among rational, complex oriented E∞ rings [BR14, 6.1]. Further, Walker studied orientations for the case of p-adic K-theory and the Todd genus [Wal08], and M¨ollers studied orientations in the case of K(1)-local spectra [M¨ol10]. Both gave proofs that Ando’s condition for H∞ orientations was also sufficient to produce E∞ orientations. The goal of this paper is to apply work by Arone–Lesh [AL07] to extend this procedure, giving an inductive approach to the construction of E∞ orienta- tions. Before getting into details, we will describe the motivation for this construction.

As an E∞ ring spectrum, the fact that MU is a Thom spectrum gives it a universal property. There is a map of infinite loop spaces U → GL1(S) from the infinite unitary group to the space of self-equivalences of the sphere spectrum S, and MU is universal among E∞ ring spectra E with a chosen nullhomotopy of the map of infinite loop spaces U → GL1(S) → GL1(E) [May77, §V]. We can recast this using the language of Picard groups. We consider two nat- ural functors: one sends a complex vector space V to the spectrum Σ∞SV , which has an inverse under the smash product; the second sends a smash- invertible spectrum I to a smash-invertible MU-module MU ∧ I. On re- stricting to the subcategory of weak equivalences and applying classifying spaces, we obtain maps of E∞ spaces

BU(m) → Z × BGL1(S) → Z × BGL1(MU). m a Here the latter two are the Picard spaces Pic(S) and Pic(MU) [MS16, 2.2.1]. Passing through an infinite loop space machine, we obtain a sequence of maps

3 of spectra ku → pic(S) → pic(MU), where ku is the connective complex K-theory spectrum. The universal prop- erty of MU can then be rephrased: the spectrum MU is universal among E∞ ring spectra E equipped with a coherently commutative diagram

ku / pic(S)

  HZ / pic(E).

(More concretely, this asks for a map HZ → pic(E) and a chosen homotopy between the two composites.) This allows us to exploit Arone–Lesh’s sequence of spectra interpolating be- tween ku and HZ, giving us an inductive sequence of obstructions to E∞ orientations.

Theorem 1. There exists a filtration of MU by E∞ Thom spectra

S → MX1 → MX2 → MX3 →···→ MU with the following properties.

1. The map hocolim MXi → MU is an equivalence.

2. There is a canonical complex orientation of MX1 such that, for all E∞

ring spectra E, the space MapE∞ (MX1, E) is homotopy equivalent to the space of ordinary complex orientations of E.

3. For all m> 0 and all maps of E∞ ring spectra MXm−1 → E, the space of extensions to a map of E∞ ring spectra MXm → E is a homotopy pullback diagram of the form

/ MapE∞ (MXm, E) MapE∞ (MXm−1, E)

  / {∗} Map∗(Fm, Pic(E))

for a certain fixed space Fm, where Pic(E) is the classifying space of the category of smash-invertible E-modules.

4 More specifically, given an E∞ map MXm−1 → E, there is an E- module Thom spectrum MEξ classified by a map ξ : Fm → Pic(E). An extension to an E∞ map MXm → E exists if and only if there is an 2m + orientation MEξ → E ∧ S in the sense of [ABG ], and the space of extensions is naturally equivalent to the space Or(ξ) of orientations.

4. The map MXm−1 → MXm is a rational equivalence if m> 1, a p-local equivalence if m is not a power of p, and a K(n)-local equivalence if m>pn.

In particular, the spectrum MX1 will be a universal complex oriented E∞ ring spectrum described by Baker–Richter [BR14].

The suspended spaces Fm are explicitly described by [AL07] as being derived ⋄ L 2m orbit spectra (Lm) ∧U(m) S . Here Lm is the nerve of the (topologized) poset of proper direct-sum decompositions of Cm, S2m is the one-point com- pactification of Cm, and ⋄ denotes unreduced suspension. Alternatively, the space Fm can be described as the homotopy cofiber of the map of Thom spaces γm γm (Lm ×U(m) EU(m)) → BU(m) for the universal bundle γm.

In the particular case of an E(1)-local E∞ ring spectrum E, such as a form of K-theory [LN14, Appendix A], this will allows us to verify that Ando’s criterion is both necessary and sufficient if E∗ is torsion-free. At higher chromatic levels there are expected to be secondary and higher obstructions involving relations between power operations. Remark 2. Rognes had previously constructed a similar filtration on algebraic K-theory spectra [Rog92], further examined in the case of complex K-theory in [AL10]. This filtration gives rise to a sequence of spectra interpolating the map ∗ → ku rather than ku → HZ. On taking Thom spectra of the resulting infinite loop maps to Z×BU, the result should be a construction of the periodic complex bordism spectrum MUP with very similar properties but slightly different subquotients, relevant to a more rigid orientation theory for 2-periodic spectra.

The authors would like to thank Greg Arone and Kathryn Lesh for discus- sions related to this material, to Andrew Baker, Eric Peterson, and Nathaniel

5 Stapleton for their comments, and to Jeremy Hahn for locating an error in the previous version of this paper.

1 The filtration of connective K-theory

In this section, we will give short background on the results that we require from [AL07, 3.9, 8.3, 9.4, 9.6, 11.3].

Proposition 3. There exists a sequence of maps of E∞ spaces

B0 → B1 → B2 → B3 →··· with the following properties.

1. The space B∞ = hocolim Bm is equivalent to the discrete E∞ space N, and the induced maps π0Bm → N are isomorphisms.

2. The space B0 is the nerve BU(n) of a skeleton of the category of finite-dimensional vector spaces and isomorphisms, with the E∞ struc- ture induced by direct sum. `

3. Let P denote the functor taking a space X to the free E∞ space on X [May72, 3.5], with the homotopy type

n P(X)= (X )hΣn . n≥0 a For each m> 0, there is a homotopy pushout diagram of E∞ spaces

/ P(Fm) / Bm−1

  / P(∗) / Bm,

where Fm is the path component of Bm−1 mapping to m ∈ N.

4. The map Fm → ∗ is an isomorphism in rational homology if m > 1, an isomorphism in p-local homology if m is not a power of p, and an isomorphism in K(n)-homology if m>pn.

6 ∞ 5. The spectrum Σ Fm is (2m − 1)-connnected.

The explicit description allows analysis of the filtration quotients using [AL07, 2.5]. The space F1 is the path component BU(1) ⊂ BU(n). We then get a homotopy commutative diagram ` / B(Σp ≀ U(1)) / BU(p) (1) PPP PPP PPP PP'  P(BU(1)) / BU(n) `    / / BΣp / P(∗) / B1 with the top map induced by the inclusion of the monomial matrices in U(p) and the left map induced by the projection Σp ≀ U(1) → Σp. The homotopy pushout of the subdiagram o / BΣp o B(Σp ≀ U(1)) / BU(p) (2) maps to Fp by a p-local homotopy equivalence. Arone–Lesh then apply an infinite loop space machine to the sequence of Proposition 3, with the following result. Corollary 4. There exists a sequence of connective spectra

b0 → b1 → b2 → b3 →··· (3) with the following properties.

1. The spectrum b∞ = hocolim bm is equivalent to the spectrum HZ, and the induced maps π0bn → Z are isomorphisms.

2. The spectrum b0 is the complex K-theory spectrum ku. 3. For each m> 0, there is a homotopy pushout diagram

∞ / Σ (Fm)+ / bm−1 (4)

  ∞ 0 / Σ S / bm.

7 4. The map bm−1 → bm is an isomorphism in rational homology if m> 1, an isomorphism in p-local homology if m is not a power of p, and an isomorphism in K(n)-homology if m>pn.

∞ 5. The homotopy fiber Σ Fm of the map bm−1 → bm is (2m−1)-connected.

2 The filtration of BU

Definition 5. For each m ≥ 0, let xm be the homotopy fiber hofib(ku → bm) ∞ of the maps from equation (3), and let Xm = Ω xm.

This gives rise to a sequence of maps

∗ ≃ x0 → x1 → x2 → x3 →··· , (5) with homotopy colimit bu ≃ Σ2ku by Bott periodicity. For each m > 0, diagram (4) and the octahedral axiom imply that the homotopy fiber of ∞ xm−1 → xm is equivalent to the desuspension ΩΣ Fm of the reduced sus- pension spectrum. Applying Ω∞ to the filtration of equation (5), we obtain a filtration of BU by infinite loop spaces:

∗→ X1 → X2 → X3 →··· (6)

∞+1 ∞ The homotopy fiber of Xm−1 → Xm is the space ΩQFm = Ω Σ Fm. In order to analyze the effect of these maps in K(n)-local homology, we will require some preliminary results. Lemma 6. Suppose x → y → z is a fiber sequence of spectra, Ω∞x is K(n)- locally trivial, and x is connective. Then the map Ω∞y → Ω∞z induces an isomorphism on K(n)-homology.

∞ ∞ Proof. By assumption π0y → π0z is surjective, so the map Ω y → Ω z is a principal fibration whose fiber over any point is Ω∞x. Applying the (nat- ural) generalized Atiyah–Hirzebruch spectral sequence, we obtain a spectral sequence ∞ ∞ ∞ H∗(Ω z; K(n)∗(Ω x)) ⇒ K(n)∗(Ω y),

8 where the E2-term may be homology with coefficients in a local coefficient system. By assumption, the edge morphism to the Atiyah–Hirzebruch spec- tral sequence ∞ ∞ H∗(Ω z; K(n)∗) ⇒ K(n)∗(Ω z) ∞ is an isomorphism on E2-terms, so it converges to an isomorphism K(n)∗(Ω y) → ∞ K(n)∗(Ω z).

Proposition 7. Let W be a based space whose suspension spectrum is at least k-connected, and define N be the family of spectra T such that Ω∞(T ∧ W ) is K(n)-acyclic. The family N has the following properties:

1. N is closed under finite wedges.

2. N is closed under filtered homotopy colimits.

3. Suppose T ′ → T → T ′′ is a fiber sequence such that T ′ is a (−k − 1)- connected spectrum in N . Then T is in N if and only if T ′′ is in N .

4. If Hk+1W is torsion, N contains the Eilenberg–Mac Lane spectrum Σn−kHA for any abelian group A. e 5. If W is K(n)-locally trivial, N contains S0.

6. If W is K(n)-locally trivial and Hk+1W is torsion, N contains all (n − k − 1)-connected spectra. e Remark 8. In particular, since Σ∞W is k-connected the assumption on n−k+1 Hk+1W holds automatically with k replaced by (k−1). Therefore, Σ HA is in N for any A, and if W is K(n)-acyclic all (n − k)-connected spectra are in N .

Proof. We will prove these items individually.

1. There is a weak equivalence

N ∞ N ∞ Ω (∨i=1Ti ∧ W ) → Ω (Ti ∧ W ), i=1 Y

9 and so this follows from Morava K-theory’s K¨unneth formula

K(n)∗(X × Y ) ∼= K(n)∗X ⊗ K(n)∗Y. K(n)∗

2. The functor Ω∞(T ∧ W ) preserves filtered homotopy colimits in T , and so there is an isomorphism

lim K(n) (Ω∞(T ∧ W )) ∼ K(n) (Ω∞((hocolim T ) ∧ W )). −→ ∗ α = ∗ α

Therefore, hocolim Tα is in N if the Tα are. 3. The spectrum T ′ ∧ W is connective, and so the result follows from a direct application of Lemma 6.

4. The spectrum Σn−kHZ ∧ W is an n-connected generalized Eilenberg– Mac Lane spectrum, and so there is a weak equivalance

∞ ∞ n−k ∼ Ω (Σ HZ ∧ W ) −→ K(Hk−n+iW, i). i=n+1 Y e By the work of Ravenel–Wilson [RW80], K(n)∗K(A, i) is trivial if i> n + 1 or if i = n + 1 and A is a torsion abelian group, and so by the K¨unneth formula Σn−kHZ is in N . Applying item 1, we find Σn−kH(ZN ) is in N ; applying item 2, we find Σn−kHF is in N whenever F is free abelian; applying item 3 to the fiber sequence

Σn−kHR → Σn−kHF → Σn−kHA

associated to a free resolution 0 → R → F → A → 0, we find that Σn−kHA is in N .

5. The Snaith splitting [Sna74] shows that we have a decomposition

∞ ∞ ∧ k Σ (QW )+ ≃ (Σ W )hΣk . k≥ _0 The spectra Σ∞W ∧ k →∗ are K(n)-locally trivial for k > 0, and so the same is true of the homotopy orbit spectra.

10 (Note that Snaith splitting is required to deduce this equivalence on the level of Morava K-theory because the equivalence does not hold on the level of mod-p homology. In particular, the K(n)-homology of a smash-power of Z is not a functor of the K(n)-homology of Z.)

6. First suppose that T is connective. By items 5 and 3, N contains any sphere Si for i ≥ 0. By items 1 and 2, N contains any wedge ∨Si for i ≥ 0. By item 3, induction on the dimension shows that N contains any connective finite-dimensional CW-spectrum. By item 2, N then contains any connective spectrum. We can now prove the general case. Suppose T is (n−k −1)-connected with n − k < 0, and consider the following portion of the Whitehead tower of T :

T [0, ∞) / T [−1, ∞) / ... / T [n − k +1, ∞) / T

   −1 n−k+1 n−k Σ Hπ−1T Σ Hπn−k+1T Σ Hπn−kT

We have just shown that T [0, ∞) is in N because it is connective, and i the spectra Σ HπiT are in N for n − k ≤ i ≤ −1 by item 4. By inductively applying item 3 we find that T is in N .

Proposition 9. The natural map ΩQFm →∗ is a rational homology equiva- lence for m> 1,a p-local equivalence for m not a power of p, anda K(n)-local equivalence if m>pn.

Proof. When m = 1 there is nothing to show. When m> 1 the suspension spectrum of Fm is k-connected for some k ≥ 2m − 1. The space Fm is also rationally trivial and p-locally trivial unless m is a power of p, so the rational homology and homotopy groups are always torsion and have p-torsion only ∞ when m is a power of p; hence the same is true for both Σ Fm and ΩQFm. If m>pn then

n − k − 1 ≤ n − (2m − 1) − 1 < n − 2pn ≤ 1 − 2p< −2, so S−1 is (n − k − 1)-connected. By Proposition 7 item 6, we then find that ∞ −1 ΩQFm = Ω (S ∧ Fm) is K(n)-locally trivial.

11 3 Decomposition of MU

Definition 10. For each m ≥ 0, let MXm be the Thom spectrum of the infinite loop map Xm → BU.

From the sequence (6) of infinite loop spaces over BU, we obtain a filtration of MU by E∞ ring spectra:

S → MX1 → MX2 → MX3 →··· (7)

Proposition 11. For any associative MU-algebra E such that ΩQFm → ∗ is an E∗-isomorphism, the map MXm−1 → MXm induces an isomorphism in E-homology.

Proof. After smashing with MU, the Thom diagonal makes the sequence of equation (7) equivalent to the sequence of MU-algebras

MU → MU[X1] → MU[X2] →···→ MU[BU], where for an E∞ space M we define MU[M]tobe E∞ ring spectrum MU ∧ M+.

The fiber sequence ΩQFm → Xm−1 → Xm of E∞ spaces implies that there are equivalences

MU ∧ (MU ∧ MXm−1) ≃ MU ∧ MXm. MU[ΩQFm]

Smashing this identification over MU with E translates this into an identity

E ∧ (E ∧ MXm−1) ≃ E ∧ MXm. E[ΩQFm]

Since the natural map E[ΩQFm] → E is an equivalence by assumption, the result follows.

By work of Lazarev [Laz01], K(n) admits the structure of an associative MU-algebra and so we can specialize this result to the case where E is a Morava K-theory. Combined with Proposition 9, this gives the following result.

12 Corollary 12. The map MXm−1 → MXm is a rational equivalence for m > 1, a p-local equivalence for m not a power of p, and a K(n)-local equivalence if m>pn.

In particular, we have the following equivalences:

(MX1)Q ≃ MUQ

LK(n)MXpn ≃ LK(n)MU

LE(n)MXpn ≃ LE(n)MU

Remark 13. This filtration on MU relies only on the existence of the map BU(n) → N of E∞ spaces. In particular, this construction is naturally equivariant for the action of the cyclic group C2, determines an equivariant ` filtration of the Real K-theory spectrum, and a sequence of C2-equivariant E∞ Thom spectra filtering the Real bordism spectrum MUR. However, the K(n)-local properties of this filtration appear to be less straightforward.

4 Picard groups

As described in the introduction, for an E∞ ring spectrum E we let Pic(E) be the nerve of the symmetric monoidal category of smash-invertible E- modules and weak equivalences [HMS94, MS16]. The symmetric monoidal structure makes Pic(E) into a grouplike E∞ space, and we write pic(E) for the associated spectrum. The 0-connected cover of pic(E) is bgl1(E). As in [ABG+], a map ξ : X → Pic(E) over the path component of an in- vertible E-module Eζ parametrizes families of E-modules over X with fibers ζ equivalent to E , and there is an associated E-module Thom spectrum MEξ. (Technically, to apply the results of [ABG+] we first need to smash with the −ζ element E ∈ π0Pic(E) to move the target to BGL1(E).) The functor sending a complex vector space to the suspension spectrum of its one-point compactification gives a map of E∞ spaces BU(m) → Pic(S), and the associated map of spectra is a map ku → pic(S). `

The space MapE∞ (MXm, E) is naturally equivalent to the space of nullhomo- + topies of the composite map xm → bu → bgl1(E) [AHR, ABG ]. However,

13 the spectra xm are 0-connected, so this is equivalent to the space of extensions in the diagram ku / pic(S)

  / bm / pic(E).

If we have already fixed an extension bm−1 → pic(E), the pushout diagram (4) expresses the space of compatible extensions to bm as the space of commuta- tive diagrams / Fm / Bm−1 (8)

  / CFm / Pic(E).

We write ξ for the diagonal composite Fm → Pic(E) in this diagram.

Proposition 14. Given an extension of ku → pic(E) to a map bm−1 → pic(E), the space of extensions to a map bm → pic(E) is equivalent to the space Or(ξ) of orientations of the E-module Thom spectrum MEξ over Fm.

Proof. We must show that the space of homotopies from ξ to a constant map is equivalent to the space Or(ξ) of orientations: maps of E-modules 2m MEξ → E ∧ S which restrict to an equivalence on Thom spectra at each point. In our case, we may choose a basepoint ∗ ∈ BU(m) classifying the vector m bundle C → ∗, whose image ∗ → Fm → Pic(E) corresponds to the E- module E ∧ S2m. We construct the following diagram of pullback squares.

Or(ξ) / {ξ}

  / / 2m Map∗(CFm, Pic(E)) Map∗(Fm, Pic(E)) {E ∧ S }

   / / Map(CFm, Pic(E)) / Map(Fm, Pic(E)) / Map(∗, Pic(E))

Here the bottom map, of necessity, lands in the path component of the E- module E ∧ S2m, and the upper-left pullback is the space Or(ξ) because

14 Fm is connected. Therefore, given this map ξ : Fm → Pic(E), the space of extensions is equivalent to the space of orientations of the E-module Thom spectrum MEξ.

5 Orientation towers

The space of E∞ orientations MU → E can now be expressed as the homo- topy limit of the tower

···→ MapE∞ (MX3, E) → MapE∞ (MX2, E) → MapE∞ (MX1, E) →∗.

The description of the space of extension diagrams from equation (8) is equiv- alent to a homotopy pullback square

/ MapE∞ (MXm, E) MapE∞ (MXm−1, E)

 2m  E∧S / {∗} Map∗(Fm, Pic(E)), where the bottom arrow classifies a constant map to the component of E∧S2m in Pic(E). The space of lifts is the space of orientations of the Thom spectrum on Fm, and so the unique obstruction to a lifting is the existence of a Thom class.

When m = 1, the space MapE∞ (MX1, E) is the space of orientations of the Thom spectrum classified by the composite

BU(1) → BU(n) → Pic(E). a More specifically, the Thom spectrum of this composite is E ∧ MU(1) ≃ E ∧ BU(1). Orientations of this are classical complex orientations: the space of orientations of this Thom spectrum is the space of maps of E-modules c1 : E∧ BU(1) → E ∧ S2 which restrict to the identity map of E ∧ S2. Therefore,

MapE∞ (MX1, E) is naturally the space of ordinary complex orientations of E.

15 6 Symmetric power operations

In order to study p-local orientations by MXp, we will need to recall the construction of power operations. Associated to a complex vector bundle ξ → X, we have the Thom space Th(ξ). This is functorial in maps of vector bundles which are fiberwise in- jections, and for the exterior Whitney sum ⊞ there is a natural isomorphism Th(ξ ⊞ ξ′) ∼= Th(ξ) ∧ Th(ξ′) that is part of a strong monoidal structure on Th. Definition 15. We define the following symmetric power functors:

× ×m Pm(X)=(X )hΣm for X a space. ∧ ∧m Pm(X)=(X )hΣm for X a based space. ∧E ∧E m Pm (X)=(X )hΣm for X an E-module. For any of the symmetric monoidal structures ? above, we will write

? ?m ? Dm : X → Pm(X) and ? ? ? ? ∆m : Pm(X ? Y ) → Pm(X) ? Pm(Y ) for the associated natural transformations.

For a vector bundle ξ → X, there is a natural vector bundle structure on the × × map Pm(ξ) → Pm(X), and we have a pullback diagram of vector bundles

× ⊞m Dm / × ξ Pm(ξ)

  ×m / × X × Pm(X). Dm Proposition 16. There are natural isomorphisms: × ∼ ∧ Th(Pmξ) = PmTh(ξ)

E ∧ Th(ξ) ∼= ME(ξ) ∧ ∼ ∧E E ∧ Pm(Y ) = Pm (E ∧ Y )

16 Definition 17. Suppose E has a chosen complex orientation u, and let ε be the trivial complex vector bundle over a point. We write

dim(ξ) tu(ξ): MEξ → ME(ε ) for the natural E-module complex orientation.

∼ 2 The map tu(ε) is the identity map of ME(ε) = E ∧ S , and tu(γ1) = c1 for the tautological bundle γ1 → BU(1). Orientations commute with exterior sum: the strong monoidal structure of Th gives us an identification

′ ′ tu(ξ ⊞ ξ ) ∼= tu(ξ) ∧E tu(ξ ).

Naturality of these orientations in pullback diagrams holds: for any map f : X → Y and vector bundle ξ → Y we have

∗ tu(f ξ)= tu(ξ) ◦ ME(f).

⊞m × × In particular, this implies that tu(ξ )= tu(Pmξ) ◦ ME(Dm).

× Definition 18. Write ρm for the vector bundle Pm(ε) on BΣm, associated m to the permutation representation of Σm on C .

∧E For this vector bundle, the naturality of Dm implies that the Thom class ⊞k ∧E tu(ρm ) ◦ Dm is the identity map. The identities above allow us to verify several relations between Thom classes.

Proposition 19. For a complex vector bundle ξ → X and a point i: ∗→ X with a chosen lift to i: ε → ξ, we have the following.

⊞m × ∧E tu(ξ )= tu(Pm(ξ)) ◦ Dm

⊞m ⊕ dim(ξ) ∧E ∧E tu(ξ )= tu(ρm ) ◦ Pm (tu(ξ)) ◦ Dm

⊕ dim(ξ) × ∧E tu(ρm )= tu(Pm(ξ)) ◦ Pm (MEi)

⊕ dim(ξ) ⊕ dim(ξ) ∧E ∧E tu(ρm )= tu(ρm ) ◦ Pm (tu(ξ)) ◦ Pm (MEi)

Corollary 20. For a complex vector bundle ξ → X, the map tu(ρm) ◦ ∧E × × Pm (tu(ξ)) is an orientation of the Thom spectrum ME(Pm(ξ)) over PmX × ×m which coincides with tu(Pm(ξ)) after restriction to X or BΣm.

17 7 Power operations

Assume that p is a fixed prime and E is an E∞ ring spectrum with a chosen complex orientation u. In this section we will recall power operations on even- degree cohomology classes [Rez]; in the case E = MU these were constructed by tom Dieck and Quillen, and used by Ando in his characterization of H∞ structures [And00].

From here on, we will write ρ = ρp for the permutation representation of Σp.

Definition 21. For an E-module spectrum M, we define

2k ∧E 2pk Pu :[M, E ∧ S ]E → [Pp M, E ∧ S ]E

⊕k ∧E by the formula Pu(α)= tu(ρ ) ◦ Pp (α).

These power operations satisfy a multiplication formula. For E-module spec- tra M and N with maps α ∈ [M, E ∧ S2k] and β ∈ [N, E ∧ S2l], we can form

2(k+l) α ∧E β ∈ [M ∧E N, E ∧ S ].

Then there is a natural identity

∧E Pu(α ∧E β) ◦ ∆p = Pu(α) ∧E Pu(β).

These power operations also depend on u, except when k = 0 where they agree with the ordinary extended power construction. Remark 22. While the formula for the power operations is given using even spheres, it implicitly relies on a fixed identification of S2k with the one-point compactification of a complex vector space.

Definition 23. For a complex vector bundle ξ → X, let j : BΣp × X → × ⊠ Pp (X) be the diagonal, and let ρ ξ → BΣp × X be the external tensor ∗ × vector bundle j Pp (ξ). We define

2k 2pk Pu : E (Th(ξ)) → E (Th(ρ ⊠ ξ)) by the formula Pu(α)= Pu(α) ◦ ME(j).

18 Definition 24. For a p-local, complex orientable multiplicative cohomology ∗ theory F , the transfer ideal Itr ⊂ F (BΣp) is the image of the transfer map ∗ ∗ F → F (BΣp), generated by the image of 1 under the transfer. For any Y ∗ equipped with a chosen map to BΣp, we also write Itr for the ideal of F (Y ) generated by the image of Itr.

The natural transformations Pu are multiplicative but not additive, instead satisfying a Cartan formula. The terms in the Cartan formula which obstruct additivity are transfers from the cohomology of proper subgroups of the form 2∗ Σk × Σp−k ⊂ Σp. If E is p-local, in the evenly-graded ring E (Th(ρ ⊠ ξ)) the mixed terms in the Cartan formula are contained inside the transfer ideal 2∗ Itr · E (Th(ρ ⊠ ξ)).

Proposition 25. The maps Pu reduce to natural maps

2∗ 2p∗ Ψu : E (Th(ξ)) → E (Th(ρ ⊠ ξ))/Itr that are additive and take any Thom class for ξ to a Thom class for ρ⊠ξ. The 2∗ maps Ψu are multiplicative, in the sense that for elements α ∈ E (Th(ξ)) 2∗ ′ and β ∈ E (Th(ξ )) we have Ψu(αβ)=Ψu(α)Ψu(β).

8 Cohomology calculations

In this section, we fix a p-local, complex orientable multiplicative cohomology theory F . Choosing a complex orientation of F , we use G to denote the as- ∗ sociated formal group law over F , and [n]G(x) the power series representing the associated n-fold sum (x +G x +G ··· +G x).

Proposition 26. The restriction map

∗ ∗ p ∗ F (B(Σp ≀ U(1))) → F (BU(1) ) × F (BΣp × BU(1)) is injective.

We will discuss a proof of this result that requires more multiplicative struc- ture from E but applies to a wider variety of objects than BU(1) in Sec- tion 10.

19 Proof. Writing B(Σp × U(1)) → B(Σp ≀ U(1)) as a map of homotopy orbit p spaces BU(1)hΣp → (BU(1) )hΣp , we obtain a diagram of function spectra

∼ / p hΣp F (B(Σp ≀ U(1)), F ) / F (BU(1) , F )

  / hΣp F (B(Σp × U(1)), F ) ∼ F (BU(1), F ) .

Therefore, the map on cohomology is the abutment of a map of homotopy fixed-point spectral sequences:

s t p + t+s H (Σp, F (BU(1) )) 3 F (B(Σp ≀ U(1)))

  s t + t+s H (Σp, F (BU(1))) 3 F (B(Σp × U(1)))

∗ ∗ ∗ p Σp The composite F (BU(p)) → F (B(Σp ≀ U(1))) → F (BU(1) ) is an iso- morphism. The latter map is the edge morphism in the above spectral se- quence, and so the line s = 0 consists of permanent cycles.

∗ p As a module acted on by the group Cp ⊂ Σp, the ring F (BU(1) ) ∼= ∗ ∗ F Jα1,...,αpK is a direct sum of two submodules: the subring F JcpK gen- k erated by the monomials ( αi) , and a free Cp-module with no higher co- homology. Therefore, for s> 0 the map Q s ∗ s ∗ p H (Σp, F JcpK) → H (Σp, F (BU(1) ))

∗ ∗ p ∗ is an isomorphism. The composite F JcpK → F (BU(1) ) → F (BU(1)) induces an injection on cohomology. The above spectral sequences are, in positive cohomological degree, the tensor products of this injective map of groups (which consist of permanent cycles) with the cohomology spectral ∗ sequence for F (BΣp), and so converge to an injective map.

Corollary 27. For any complex vector bundle ξ → B(Σp ≀ U(1)), two Thom classes for ξ are equivalent if and only if their restrictions to BU(1)p and BΣp × BU(1) are equivalent.

Proof. The product of the restriction maps on the F -cohomology of Thom spaces is injective by naturality of the Thom isomorphism.

20 Proposition 28. If F ∗ is torsion-free, the map

∗ ∗ ∗ F (BΣp × BU(1)) → F (BU(1)) × F (BΣp × BU(1))/Itr is injective.

(This is similar to results from [HKR00], though here we do not assume that the coefficient ring F ∗ is local.)

Proof. The natural K¨unneth isomorphisms on the skeleta CPk ⊂ BU(1) take the form ∗ k ∗ ∗ k F (BΣp × CP ) ∼= F (BΣp) ⊗F ∗ F (CP ). This inverse system in k also satisfies the Mittag–Leffler condition, and so ∗ ∗ ∗ it suffices to show that the map F (BΣp) → F × F (BΣp)/Itr is injective. As the cyclic group Cp ⊂ BΣp has index relatively prime to p, the left-hand map is injective in the commutative diagram

F ∗(BΣ ) / F ∗ × F ∗(BΣ )/I  _ p p tr

  ∗ / ∗ ∗ F (BCp) / F × F (BCp)/Itr.

It therefore suffices to prove that the bottom map is injective.

∗ As p is not a zero divisor in F , the p-series [p]G(x) is not a zero divi- ∗ ∗ sor in F (BU(1)), and so the cohomology ring F (BCp) is the quotient ∗ ∗ ∗ F Jc1K /[p]G(c1) [HMS94]. The kernel of the map F (BCp) → F is gen- erated by c1, while the transfer ideal is generated by the divided p-series ∗ ∗ hpiG(c1) = [p]G(c1)/c1 by naturality of the map MU → F [Qui71, 4.2]. The intersection of the ideals (c1) and (hpiG(c1)) in the power series ring consists of elements g(c1) ·hpiG(c1) such that the constant coefficient of g is ∗ annihilated by the constant coefficient p of hpiG(c1). As F is torsion-free, this ideal is generated by [p]G(c1).

Corollary 29. For any complex vector bundle ξ → BΣp × BU(1), two ori- entations for ξ are equivalent if and only if their restrictions to BU(1) are ∗ ∗ ∗ equal and their images in F (BΣp)/Itr ⊗F (BΣp) F (Th(ξ)) are equal.

21 9 Orientations by MXp

∗ In this section, we fix a p-local E∞ ring spectrum E such that E is torsion- free, together with a complex orientation u defined by a map MX1 → E. (We will continue to write G for the associated formal group law.) In this section we will analyze the obstruction to p-local maps from MXp.

As in Section 5, the space of extensions of the complex orientation to an E∞ ring map MXp → E is the space of orientations of the Thom spectrum over Fp. The homotopy pushout diagram for Fp from equation (2) expresses the map Fp → B1 → Pic(E) as a coherently commutative diagram

/ B(Σp ≀ U(1)) / BU(p)

γp   / BΣp ρ Pic(E).

From diagram (1), the map BU(p) → Pic(E) classifies the Thom spectrum MEγp = E ∧ Mγp associated to the tautological bundle γp of BU(p), while the map BΣp → Pic(E) classifies the bundle MEρ associated to the regular representation ρ: Σp → U(p). An orientation of the resulting Thom spec- trum over Fp exists if and only if there are orientations of MEγp and MEρ whose restrictions to B(Σp ≀ U(1)) agree.

Proposition 30. There exists an orientation of the Thom spectrum over Fp if and only if the orientations tu(ρ) and tu(γp) have the same restriction to B(Σp ≀ U(1)), or equivalently if

∧E tu(ρ) ◦ Pp (γ1)= tu(Pp(γ1)).

Proof. The “if” direction is clear. In the other direction, we start by assuming that we have some pair of orientations whose restrictions agree.

0 × Any orientation of MEρ is of the form a · tu(ρ) for some a ∈ E (BΣp) , and similarly any orientation of MEγp is of the form b · tu(γp) for some b ∈ 0 × ∧E E (BU(p)) . These restrict to a·tu(ρ)◦Pp (γ1) and b·tu(Pp(γ1)) respectively. By Corollary 20, the restrictions of these orientations to BU(1)p are ǫ(a) · ⊞p ⊞b 0 × tu(γ1 ) and b · tu(γ1 ), where ǫ(a) is the natural restriction of a to (E ) and

22 we identify b with its image under the injection E0(BU(p)) → E0(BU(1)p). Similarly, the restrictions of these orientations to BΣp are a · tu(ρ) and ǫ(b) · tu(ρ) respectively. For these to be equal as needed, both a and b must be in the image of (E0)× and equal. Changing the orientation by multiplying by a−1 then gives the desired result.

Combining this with Corollary 20 and Corollary 29, we find the following.

Proposition 31. There exists an orientation of the Thom spectrum over Fp if and only if the orientations tu(ρ) and tu(γp) define the same generating ∗ class after first restricting to BΣp ×BU(1) and then tensoring over F (BΣp) ∗ with F (BΣp)/Itr.

We recall for the following that, if E∗ is torsion-free, we have an isomorphism

× × ∗ ∗ Fp ∗ Fp E (BΣp) ∼= E (BCp) =(E JzK /[p]G(z)) ,

× where the action of Fp is by z 7→ [i]G[z].

∗ Theorem 32. If E is an E∞ ring spectrum such that E is p-local and torsion-free, an E∞ orientation MX1 → E extends to an E∞ orientation MXp → E if and only if the power operation Ψu satisfies the Ando criterion: we must have p−1

Ψu(c1)= (c1 +G [i]G(z)) i=0 Y 2pk in the ring E (Th(ρ ⊠ γ1))/Itr.

Proof. It is necessary and sufficient, by Proposition 31, to know that the ∧E restrictions of tu(ρ) ◦ Pp (γ1) and tu(Pp(γ1)) to this target ring are equal. By definition, the former restricts to Ψu(tuγ1)=Ψu(c1). The latter restricts to tu(ρ ⊠ γ1), and so this formula for the Thom class follows from the splitting principle.

23 10 Cohomology monomorphisms for E-theory

In this section we will show the following result, which is closely related to Proposition 26 when X = BU(1)+.

Proposition 33. Let E be a p-local, complex orientable E∞ ring spectrum whose coefficient ring has no p-torsion, and let X be a based space with p-fold (p) smash power X such that E∗X is a direct sum of (unshifted) copies of E∗. Then the restriction map

∗ (p) ∗ (p) ∗ E (XhCp ) → E (X ) × E ((BCp)+ ∧ X) is a monomorphism.

For instance, this is satisfied when E is Morava E-theory and X is of finite type with Z(p)-homology concentrated in even degrees. This allows us to remove the finite type hypothesis from the cohomology theory in [BMMS86, VIII.7.3] so that it applies to Morava E-theory (e.g. see [And95, 4.4.2], [AHS04, proof of 6.1]). We would like to thank Eric Peterson and Nathaniel Stapleton for bringing this to our attention.

Proof. Write M = F (Σ∞X(p), E) for the function spectrum, which has an ∞ action of Cp from the source, and N = F (Σ X, E), with the trivial action of Cp. The homotopy fixed-point map

M hCp → M,

∗ (p) ∗ (p) on homotopy groups, becomes the map E (XhCp ) → E (X ). On the other hand, the map M hCp → N hCp ∗ (p) ∗ becomes the map E (XhCp ) → E ((BCp)+ ∧ X). We want to prove that these are jointly monomorphisms.

By the assumptions on X, we have that E ∧ X ≃ α E as E-modules, and (p) ∧E p so there is a Cp-equivariant equivalence E ∧ X =∼ (E ∧ X) . Using the decomposition of E ∧ X into a wedge of copiesW of E, this decomposes Cp-equivariantly as an E-module into a Cp-fixed component and a Cp-free

24 component:

∧E (p) ∼ (E ∧ X) = E ∨ (Cp)+ ∧ E . α ! β ! _ _ The spectrum M is E-dual to this, and we calculate

hCp π∗M ∼= E∗ JxK /[p]F (x) × E∗. α ! β Y Y We find that the map M hCp → M is a monomorphism on the right-hand factor. On the left-hand factor coming from the terms with trivial action, it becomes a product of projection maps

E∗ JxK /[p]F (x) → E∗ α α Y Y which send x to zero. The kernel of this consists precisely of the multiples of x. Therefore, to finish the proof, we simply need to show that the multiples of x map monomorphically into the homotopy of N hCp . We now consider, for any finite subcomplex Y ⊂ X, the natural diagram of homotopy fixed-point and Tate spectra:

F (X(p), E)hCp / F (X, E)hCp

  F (X(p), E)tCp / F (X, E)tCp

  F (Y (p), E)tCp / F (Y, E)tCp .

The upper left-hand map is the localization

−1 E∗ JxK /[p]F (x) × E∗ → x E∗ JxK /[p]F (x), α ! β α Y Y Y which is a monomorphism on the multiples of x.

25 The bottom map is a natural transformation of functors in Y , and is evidently an equivalence when Y is S0. Both of these functors take cofiber sequences of based spaces Y to fiber sequences of spectra, as follows. For a cofiber sequence Y ′ → Y → Y ′′, the smash power of Y has an equivariant filtration whose k’th associated graded consists of smash products of k copies of Y ′ and (p − k) copies of ΣY ′′; the terms other than k = 0 and k = p are acted on freely by Cp and do not contribute to the Tate spectrum. Therefore, the bottom map is an equivalence for any finite complex Y . Given our space X, we observe that

colim E∗Y ∼= E∗X ∼= ⊕αE∗ Y as Y ranges over the filtered system of finite subcomplexes of X. Therefore, for any index α the corresponding generator of E∗X lifts to E∗Y for some (p) tCp such Y . Any nonzero element in π∗F (X , E) then has nonzero restric- (p) tCp tion to π∗F (Y , E) for some Y , and hence (since the bottom map is an tCp isomorphism) has nonzero image in π∗F (X, E) . Thus, the center map is injective. Since the center map is injective and the upper-left map is injective on mul- tiples of x, the top map in the diagram is also injective on multiples of x as desired.

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